Signed ternary carry generator using threshold logic elements



Oct. 10, 1967 w. H. HANsoN SIGNED TERNARY CARRY GENERATOR USING THRESHOLD LOGIC ELEMENTS 2 Sheets-Sheet l Filed sept. 30, 1965 GROUP-l 0 mlhGRou Cum-mi *ILLCQ J l m J:

Aqu-ml Q OO GO H+ H+ ...A .dw mm mm qq qq AB AB M ,m RH Y O .MH WM mMA M ocr. 1o, 1967 w. H. HANSQN 3,346,730

SIGNED TERNARY CARRY GENERATOR USING THRESHOLD LOGIC ELEMENTS Filed Sept. 30, 1965 l y 2 Sheets-Sheet 2 l-Sl ELEMENT 2M ELEMENT 3 ELEMENT im ELEMENT qu-n+1 Bau-n+1 lNvENToR W/l. L/AM H. HANSOA/ ToRNEY g (m Hm ELEMENT m.- ELEMENT United. States Patent O ABSTRACT F THE DISCLOSURE A signed ternary carry generator utilizing multi-input ternary threshold logic elements for performing ternary arithmetical operations.

Both the conventional and signed ternary number systems, along with associated arithmetical operations, are dened and discussed in copending patent application Ser. No. 220,183, filed Aug. 29, 1962, and entitled' Data Processing System.

Conventional ternary comparator and carry' networks are generally serial in nature and consist ofl a serial arrangement of ternary carry and comparator stages such as those disclosed in a paper `by W. H. Hanson entitled Ternary Threshold Logic, published in IEEE Transactions on Electronic Computers, vol. ECilZ, pp. 191-197; June 1963. Serial networks of this type require a relatively large number of elements and logic levels t'o effect. Some further properties of ternary threshold logic are discussed in a note by R. Merrill entitled Some Properties of Ternary Threshold Logic published in IEEE Transactions on Electronic Computers, vol. EC-13; pp. 632-635; Oci tober 1964.

It has been determined that the carry, corresponding to the q( j-l) -i-z' digit-order, of two signed ternary words oftheforrnAn...Ak...A1andBn... is represented by the logical function, d

Cq i1 +i=waqo-niwiBqriln-H 1iwlAqunliwiBqo-nrwiCQ i1 It has further been determined that a comparison, through the jqth digit-order, of two ternary words of the form An Ak A1 and Bn .y Bk B1 is represented by the logical function,

Kiq;2wq.Auit2waFqit tZvwAq(i-i;+i2wiFq(i-1li y2109114qG-1i-H2w1qu-1HKQ(H) It is proposed by this invention to reduce the number of elements and logic levels necessary toV generate signals representative of signed ternary carries and ternary comY parisons by utilizing these determinations and their logical equivalents.

The invention is effected by employingk multi-input ternary threshold logic elements. There are a variety of circuits available, capable of performing the logical threshold function, which are well known in the art (such as described by R. Schauer et al. in Some Applications of Magnetic Film Parametrons as Logical D`evices,.-IRE Transactions onv Electronic Computers, vol. EC-9, pp. 315-320; September 1960) and these circuits, ofthem'- selves, do not constitute a part of this invention.

The output, Y, of 4a ternary threshold logic element having n inputs Xn Xk X1 with associated weights wIx can be represented as follows:

. wk w1, where wkl, and threshold t, where' representations of digit-orders A111 .i augend, q signal representations of digit-orders Bq,

3,346,730 Patented Oct. 10, 1967 ice The value of the ternary threshold logic function, Y, is determined as follows:

In the signed ternary'nurnber system;

Y='O in all other cases, where X k= -l-l, 0 -l and in the conventional ternary number system;

0 both as to its organization and method of operation, as

Well as additional advantages thereof, will be best understood from the following description when yread in connection with the accompany drawings, in which:

FIGURE 1 i-s a logical block diagram of a preferred embodiment of a carry generator designed in accordance with this invention; and

FIGURE 2 is a logical block diagram of a preferred embodiment of a comparator designed in accordance with this invention.

The drawings generally show block diagrams of a basic carry generator land comparator designed in accordance with the present inventionnIn thesegures each of the vlblocks representa multi-inputternaiy threshold logic elementfThe termappearingwithin each of the' blocks represents the threshold of the element; The term appearing at each input to each of the elements represents the weight accorded the associated input.

. With reference now to FIGURE 1 of the drawings, a logical block diagram of a carry generator accordingto the present invention is shown. The carry'generator, for generating n signals representative of carries, Cn

Ck C1, corresponding to an augend of the form An Ak A1 and an addend of the form Bn Bk B1, comprises m signal generating groups arranged in m logic levels. The number of groups, m, necessary to generate n signal representations of Cx1 Ck C1, is given by, n/q where;

n/q =n/g if the quotient is an integer,

=n/ q rounded lup to the next higher integer, if

the quotient is not an integer, q: p- 1 2, and pis the number of distinct input signals allowed per element. The total number of input signals allowed per element is given by, 31.

Each of the groups, j=1 m-l, receives q signal Bq(1 1)-+1 of the addend and a signal representation of Cq(1 1),the highest digit-order carry generated by the next lower order group (where, C0=0). Each of these groups generate qcarry signals, Cqj Cq(,- 1)+1, as Va function Bq m1 +r Buen-n+1 of the addend, and a signal representation of Cq(m 1) the highest order carry generated by the (m--1)th group, where r is the non-zero remainder of n/ q. If the remainder of n/ q is zero then r=q. The mth group generates r carry signals, Cn Cq(m 1)+1, as a function only of the digit-order inputs to the group and the highest order carry generated by the next lower group.

`Each of the groups, j=l m-l, comprises q multiinput ternary threshold logic elements arranged in pai-aL lel. The ith element, =1 q, of the jth group has a threshold of l1: (t/Z) (31-|1)(t=l, .and is coupled so as to receive 21'2-1-1 input signals representative of digitorders Aq(j 1)+1 Aq j 1)+1 of said augend, digit-ordeIS Bq(j 1)+1 Bq(j 1)+1 0f Said added, and Cq(j 1), the latter being representative of the highest digit-order carry generated by the next lower order group. Each of said digit-order inputs is weighted by a factor ivf- (1) (3i-1) and Cq(j 1) is weighted by a factor of w1. The ith element First group:

Second group:

Third group:

Fourth group:

jth group;

Fourteenth group:

Fifteenth group:

where j:l m-l and for each j, i=1 q.

First group:

The mth group is comprised of r multi-input ternary threshold logic elements arranged in parallel. The ith element, 1:1, r, of the mth group has a threshold of l1:(t/2) (3i+1)(t:l, and is coupled so as to re- -ceive 2l1 input signals representative of digit-orders Aq(m 1)+1 Aq(m 1)+1 of said augend, digit-orders Bq(m 1)+1 Bq(m. 1)+1 Of Sald addelld, and Cun-1 4),

the latter being representative of the highest digit-order carry generated by the next lower order group, mi--L Each of said digit-order inputs are weighted by a factor w,:(t) (3i-1) and Cq(m 1) is weighted by a factor of w1. The ith element of the mth group utilizes said digit-order and carry signals to generate a signal in accordance with the logical function:

Following are two examples of specific signed ternary carry generators designed in accordance with this invention. These examples are intended as illustrative only and are not to be construed as limiting.

Let p:5, t:l, and 11:30. Then q=2, m1=15, 11:2, w,:3i*1, and 11:1/2 (SLi-l). With these parameters given the specic carry generator is dened by the following logical functions:

Let 17:7, t=l, and '1:29. Then q=3, 111;:10, 1*:2, w1=311, and l1=11/ (3i-H). With these parameters given the specific carry generator is defined by the following logical functions:

Second group:

7' th group:

Ninth group:

Tenth group:

C29 3142953 3295A 2853285 C 21 With reference now to FIGURE 2 of the drawings, a logical block diagram of a comparator according to the present invention is shown. The comparator, for generating a signal, Kn, representative of a comparison of two ternary words of theform An Ak A1 and Bn Bk B1, comprises m multi-input ternary threshold logic elements arranged in m logic levels. The number of elements, m, necessary to generate K,l1 is given by, n/q where:

n/ q :n/ q, if the quotient is fan integer,

:n/ q rounded up to the next higher integer, if

the quotient is not an integer, q: (p-l)/2, and p is the number of distinct input signals allowed per element. The total number of input signals allowed per element is given by (2) (31)-1.

Each of the elements, j:1 m-l, receive 2q signal representations of digit-orders Aqj Aq(j 1)+1 and Bqj B(j 1)+1, of the words to be compared, A and B, and a signal representation of Kq(j 1), the signal generated by the next lower order element. Each of these elements generates a signal Kqj, representative of a comparison 'through the qi digit order. The mth element receives 2r signal representations of digit-orders An An r+1 and -Bn Bn ,+1, of the words to be compared, A and B,

and 'a signal representation of Kn the signal generated by the (m-'l)th element, where r is the non-zero remain- Ader of n/ q. If the remainder of t/ q is zero then r:q. The

First element: Second element:

where 17:1 .`r.

Control signal K0, an input to the iirstA element, determines what condition Kn will detect, in accordance with the following:

1 Kn: 1 implies A 2B K`= 1' implies A B Kn=1 implies A B Kn=0 implies A=1B 30 Following are two examples of specific ternary comparators designed in accord-ance with this invention. These examples are intended as illustrative only and are not to be construed as limiting. Y v

35 Let 17:5, t:1 and n:30. Then q:2, m:15, r:2, and 2wi=(2)(311). Withv these parameters given the Yspecific comparator is defined Abythe-following logical functions.

y Fifteenth element:

parison signals to generate a signal in accordance with the logical function:

where j:1 m-1 and for each j, i:l q.

First element: Second element:

h element:

Ninth element: Tenth element:

The number of distinct input signals allowed per element, p (or the total number of input signals allowed per element), is determined by the specific type of hardware utilized to implement the invention. The factor, t, which is utilized in determining the threshold of the various elev ments is preferably one; however any other integral value will render the invention operative.

It is understood that suitable modifications may be made in the structure as disclosed provided such modifications come within the spirit and scope of the appended claims. Having now, therefore, fully illustrated and described my invention, what I claim to be novel and desire to protect by Letters Patent is:

What is claimed is:

1. A signed ternary carry generator, for generating n signals Cn Ck C1 representative of carries corresponding to two ternary words the augend of which has the form An Ak A1 and an addend of which has the form Bn Bk B1, comprising:

m carry generating groups for generating signal representations of Cn Ck C1,

each of said carry generating groups j=l, m-l,

coupled to the next lower carry generating group, for generating q signals representative of carries Cq, Cq(j 1)+1 corresponding to digit-orders Aqj Aq(j 1)+1 of said augend and digit-orders Bqj Bq( 1)+1 of said addend comprising:

input means for receiving signal representations Of digit-Orders Aqj Aq(j 1)+1, Bq] Bq(j 1)+1, and carry signal Cq(j 1), the latter being representative of the highest digit-order carry generated by the next lower carry generating group, and a plurality of multi-input ternary threshold logic elements coupled to said input means for utilizing said digit-order and carry signals to generate q signals representative of the carries Cqj Ccmn+1 in accordance with the logical function,

said carry generating group, ]'=m, coupled to the next lower carry generating group, for generating r signals representative of the carries Cn Cq(m 1)+1 corresponding to digit-orders An Aq(m 1)+1 of said augend and digit-orders Bn Bq(m 1)+1 of said addend comprising:

input means for receiving signal representations of digit-orders An Aq(m 1)+1, Bn Bq(m 1)+1 and carry signal Cq(m 1), the latter being representative of the highest digit-order carry generated by the next lower carry generating group, and r multi-input ternary threshold input logic elelments coupled to said input means for utilizing said digit-order and carry signals to generate r signals representative of CIl Cq(m 1)+1 in accordance with the logical function,

2. A carry generator as defined in claim 1 in which each of said groups of multi-input ternary threshold logic elements are signed ternary logic elements arranged such that said carry signal representations are generated in m logic levels.

3. A carry generator as dened in claim 2 in which m is defined by the function 11/ q q=(p-l)/2, py is the number of distinct input signals per threshold element, and r is the non-zero remainder of n/ q.

4. A carry generator .as .dened in claim 3 in which each of said networks are comprised of q signed ternary threshold logic elements arranged in parallel.

5. A carry generating group, y', for generating q signals representative of Cqj Cq(j 1)+1 corresponding to digit-orders Aqj Aq(j 1)+1 of an augend of the form An Ak A1 and digit-orders Bqj Bq(5 1)+1 of an addend of the form Bn Bk B comprising:

input means for receiving signal representations of digli-Orders Aqj Aq(j 1)+1, Bqj Bq(j 1)+1, and carry signal Ccm- 1), the latter being representative of the highest digit-order carry generated by a next lower carry generating group, and

a plurality of multi-input ternary threshold logic elementsl coupled to said input means for utilizing said digit-order and carry signals to generate signals representative of Cm Cq(j 1)+1 in accordance with the logical function,

'i liw1A qu-nliwiB q in+1iw10q (i-i) where i designates a particular element in said group, w1=the Weight of the individual digit-order inputs to said particular element and li is the threshold of said particular element.

6. A carry generating group as dened in claim 5 in which said plurality of multi-input ternary threshold logic elements are signed ternary threshold logic elements arranged such that said carry signal representations are generated in one logic level.

References Cited UNITED STATES PATENTS 3,202,806 8/1965 Menne 235-175 MALCOLM A. MORRISON, Primary Exwminer.

V. SIBER, Assistant Examiner. 

1. A SIGNED TERNARY CARRY GENERATOR, FOR GENERATING N SIGNALS CN... CK... C1 REPRESENTATIVE OF CARRIES CORRESPONDING TO TWO TERNARY WORDS THE AUGEND OF WHICH HAS THE FORM AN... AK... A1 AND AN ADDEND OF WHICH HAS THE FORM BN... BK... B1, COMPRISING: M CARRY GENERATING GROUPS FOR GENERATING SIGNAL REPRESENTATIONS OF CN... CK... C1, EACH OF SAID CARRY GENERATING GROUPS J=1,... M-1, COUPLED TO THE NEXT LOWER CARRY GENERATING GROUP, FOR GENERATING Q SIGNALS REPRESENTATIVE OF CARRIES CQJ... CQ(J-1)+1 CORRESPONDING TO DIGIT-ORDERS AQJ... AQ(J-1)+1 OF SAID AUGEND AND DIGIT-ORDERS BQJ... BQ(J-1)+1 OF SAID ADDEND COMPRISING: INPUT MEANS FOR RECEIVING SIGNAL REPRESENTATIONS OF DIGIT-ORDERS AQJ... AQ7J-1)+1, BQJ... BQJ(J-1)+1, AND CARRY SIGNAL CQ(J-1), THE LATTER BEING REPRESENTATIVE OF THE HIGHEST DIGIT-ORDER CARRY GENERATED BY THE NEXT LOWER CARRY GENERATING GROUP, AND A PLURALITY OF MULTI-INPUT TERNARY THRESHOLD LOGIC ELEMENTS COUPLED TO SAID INPUT MEANS FOR UTILIZING SAID DIGIT-ORDER AND CARRY SIGNALS TO GENERATE Q SIGNALS REPRESENTATIVE OF THE CARRIES CQJ... CQ(J-1)+1 IN ACCORDANCE WITH THE LOGICAL FUNCTION, 